
Date: Sun, 06 Jun 1999 22:38:37 0700
Subject: word problem  four weights
Four Weights
When using a balance scale, weights can be placed on either side of the scale. For example, if a 10 pound weight provides a counter balance to an object and a 7 pound weight, then the object must weigh 3 pounds. What four weights can be used to weigh objects of 1, 2, 3 ... 38, 39, 40 pounds?
Hi,
There are a number of approaches running from trial and error to use of recurrences and the binomial theorem but the simplest way to look at this problem in general, i.e. to show how one can use n weights to balance the weights 1, 2, ... ,(3^{n}  1)/2, is to use what we call base 3 arithmetic. Without being too theoretical we will try to show you the algorithm of how to make a weight balance.
Recall that numbers that we commonly use such as 3456 are what we call base 10 numbers, i.e.
3456 = 3*10^{3} + 4*10^{2} + 5*10^{1} + 6.
In other words
3456 = 3*1000 + 4*100 + 5*10 + 6.
This is a unique representation in terms of powers of 10. There is nothing special about powers of 10, we could just as easily write numbers in terms of powers of 3; for example, 70 is the same as
2*3^{3} + 1*3^{2} + 2*3^{1} + 1 = 2*27 + 1*9 + 2*3 + 1.
We would say that the base 10 number 70 is 2121 in base 3, that is 70 = 2121_{3}. These 'digits' 2, 1, 2 and 1 are the coefficients of the different powers of 3 required.
Similarly in base 3,
201102_{3}^{ } 
= 
2*3^{5} + 0*3^{4} + 1*3^{3} + 1*3^{2} + 0*3^{1} + 2_{ } 

= 
2*243 + 1*27 + 1*9 +2 

= 
486 + 27 + 9 + 2 

= 
524. 
Here are the first 40 base 10 numbers and their base 3 equivalents.
Base 10  Base 3  Base 10  Base 3 
1  1 
21  210 
2  2 
22  211 
3  10 
23  212 
4  11 
24  220 
5  12 
25  221 
6  20 
26  222 
7  21 
27  1000 
8  22 
28  1001 
9  100 
29  1002 
10  101 
30  1010 
11  102 
31  1011 
12  110 
32  1012 
13  111 
33  1020 
14  112 
34  1021 
15  120 
35  1022 
16  121 
36  1100 
17  122 
37  1101 
18  200 
38  1102 
19  201 
39  1110 
20  202 
40  1111 
Suppose that you want to balance say thirtyone pounds. Imagine that you have put thirtyone pounds on each side of the balance. On the left is thirtyone pounds written in base 10 notation (31) and on the right is twelve pounds written in base 3 notation (1011_{3}). Hence on the right is 1*3^{3} + 1*3^{1} + 1 so you can balance 31 pounds with one 27 pound weight, one 3 pound weight and one 1 pound weight. In a similar fashion any weight which has only ones and zeros when written in base 3 notation is easy to balance. For example, looking at the table above, 13 pounds balances with one 9 pound weight, one 3 pound weight and one 1 pound weight.
Suppose now that you want to weigh thirtythree pounds. Thirtythree in base 3 notation is 1020_{3} but you don't have two 3 pound weights to put on the right. Put your 3 pound weight with the thirtythree pound weight on the left side of the balance and then to balance you need an extra three pounds on the right. That is on the right you need one 27 pound weight and three 3 pound weights which is equivalent to one 27 pound weight and one 9 pound weight. That is 33 + 3 balances with 1020_{3} + 10_{3} = 1100_{3}. Thus put your 1 pound weight on the left with the 33 pound weight and balance it with a 27 pound weight and a 9 pound weight.
As a second example look at seventeen pounds. Seventeen in base 3 is 122_{3}. Since there is a 2 in the first position (reading from the right) put your 1 pound weight on the left, then to balance on the right you need eighteen pounds. Eighteen is 200_{3} which is two nines so again you need to put your 9 pound weight on the left, forcing another 9 pounds on the right for a sum of twentyseven pounds. Thus 17 + 1 + 9 balances with 122_{3} + 1_{3} + 100_{3} = 1000_{3} which is 27 pounds. Thus to weight 17 pounds put the 1 and 9 pound weights on the left with the 17 pound weight and the 27 pound weight on the right.
What you are actually doing here is looking for a number which contains only zeros and ones when written in base 3, that when added to 122_{3} in base 3 results in a total that contains only zeros and ones. This number is 101_{3} since
122_{3}
+101_{3}

1000_{3}
Starting from the right, one plus two is three, but three is 10_{3} so write 0 and carry the 1. In the next column two plus the one you carried is three, i.e. 10_{3}, so again write 0 and carry the 1. Finally one plus one plus one is three which is 10_{3} so write 0 and carry the 1 for a total of 1000_{3}.
Cheers,
Penny
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